Heesch Numbers of Unmarked Polyforms

Total Page:16

File Type:pdf, Size:1020Kb

Heesch Numbers of Unmarked Polyforms Heesch Numbers of Unmarked Polyforms Craig S. Kaplan School of Computer Science, University of Waterloo, Ontario, Canada; csk@uwaterloo.ca Abstract A shape’s Heesch number is the number of layers of copies of the shape that can be placed around it without gaps or overlaps. Experimentation and exhaustive searching have turned up examples of shapes with finite Heesch numbers up to six, but nothing higher. The computational problem of classifying simple families of shapes by Heesch number can provide more experimental data to fuel our understanding of this topic. I present a technique for computing Heesch numbers of non-tiling polyforms using a SAT solver, and the results of exhaustive computation of Heesch numbers up to 19-ominoes, 17-hexes, and 24-iamonds. 1 Introduction Tiling theory is the branch of mathematics concerned with the properties of shapes that can cover the plane with no gaps or overlaps. It is a topic rich with deep results and open problems. Of course, tiling theory must occasionally venture into the study of shapes that do not tile the plane, so that we might understand those that do more completely. If a shape tiles the plane, then it must be possible to surround the shape by congruent copies of itself, leaving no part of its boundary exposed. A circle clearly cannot tile the plane, because neighbouring circles can cover at most a finite number of points on its boundary. A regular pentagon also cannot be surrounded by copies of itself: its vertices will always remain exposed. However, the converse is not true: there exist shapes that can be fully surrounded by copies of themselves, but for which no such surround can be extended to a tiling. For example, there are 108 heptominoes (shapes formed by gluing together seven squares), of which four, shown in Figure1, are known not to tile the plane. One of them contains an internal hole and can be discarded immediately. As it happens, the other three can all be surrounded. In the middle two cases, the shape and its surrounding copies are simply connected. On the right, the surrounding tiles leave behind an internal hole, and no alternative surround can eliminate that hole. There is no a priori reason why a given non-tiling shape might not be surroundable by two, three, or more layers of copies of itself. The illustrations in Figure1 provide lower bounds for the numbers of layers arXiv:2105.09438v1 [cs.CG] 20 May 2021 Figure 1: The four non-tiling heptominoes. The shape on the left has a hole and cannot be surrounded. The other three can be fully surrounded by copies, but in the rightmost shape the copies will necessarily enclose a hole. 1 Figure 2: A 23-omino that can be surrounded by two layers of copies of itself, but not more. for these shapes; that they also represent upper bounds must be proven by enumerating all possible surrounds, and showing that none of them may be further surrounded. Other shapes might permit more layers. For example, the 23-omino shown in Figure2, due to Fontaine [3], can be surrounded by two layers but not more. How far can this process be extended? A shape’s Heesch number is the number of times it can be surrounded with complete layers of congruent copies of itself (I will offer a precise definition in the next section). If the shape tiles the plane, its Heesch number is defined to be infinity. Heesch’s problem asks which positive integers are Heesch numbers; that is, for which = ¡ 0 does there exist a shape with Heesch number =? Very little is known about the solution to Heesch’s problem. Writing in 1987, Grünbaum and Shephard were not aware of any examples with finite Heesch number greater than 1 [5, Section 3.8]. After that, a few isolated examples were found with Heesch numbers up to 4 [7]. Mann and Thomas performed a systematic computer search of marked polyforms (polyominoes, polyhexes, and polyiamonds, with edges decorated with geometric matching conditions), yielding new examples and pushing the record to 5 [8]. In 2021 Bašić finally broke this record, demonstrating a figure with Heesch number 6 [1]. The study of Heesch numbers can shed light on some of the deepest problems in tiling theory. In particular, the tiling problem asks, for a given set of shapes, whether they admit at least one tiling of the plane. The tiling problem is known to be undecidable for general sets of shapes [2], but its status is open for a set consisting of a single shape (. If there were an upper bound # on finite Heesch numbers, then the tiling problem would be decidable, at least when there are only finitely many ways that two copies of ( may be adjacent [4]. The algorithm would involve trying the finitely many ways of surrounding ( with # ¸ 1 layers of copies of itself. If you succeed, then you have exceeded the maximum finite Heesch number and ( must tile the plane. If you fail, then ( evidently does not tile. To that end, more experimental data revealing which Heesch numbers are possible, even for limited classes of shapes, could be useful in understanding whether such an upper bound might exist. In this article I report on a complete enumeration of Heesch numbers of unmarked polyforms, up to 2 19-ominoes, 17-hexes, and 24-iamonds. This enumeration comprises approximately 4.16 billion non-tilers, extracted from enumerations of all free polyforms of those sizes. Respecting a slight difference of opinion among researchers, I compute two variations of Heesch numbers: one where tiles may form holes in the outermost layer, and one where a shape and all its surrounding layers must be simply connected. This enumeration does not shatter the existing records for Heesch numbers, but it does provide a store of new examples of shapes with non-trivial Heesch numbers. Some, like a 9-omino with Heesch number 2 (Figure7) and a 7-hex with Heesch number 3 (Figure8), are interesting because of the complex behaviour exhibited by relatively simple shapes. The enumeration also uncovered seven new examples with Heesch number 4. Apart from the tabulation and specific examples, the other main contribution of this work lies in the use of a SAT solver to compute Heesch numbers. Because polyominoes, polyhexes, and polyiamonds are subsets of ambient regular tilings of the plane, it is possible to reduce the geometric problem of surroundability to the logical problem of satisfiability of Boolean formulas. A SAT solver can optimize its search of the exponential space of possible solutions, avoiding the risk of “backtracking hell”. This formulation leads to a very reliable algorithm, whose performance degrades only on the rare shapes that actually have high Heesch numbers. 2 Mathematical background Although Heesch’s problem grew out of tiling theory, most of the language, techniques, and results of tiling theory are not needed within the scope of this article and will be omitted. Readers interested in the topic should consult Grünbaum and Shephard’s book [5], which remains the standard reference. In this section I will formalize the definition of a shape’s Heesch number and review marked and unmarked polyforms. 2.1 Heesch numbers Let 퐶 and ( be simple shapes in the plane, i.e., topological discs. We say that 퐶 can be surrounded by ( if there exists a set of shapes f(1,...,(=g with the following properties: 1. Each (8 is congruent to ( via a rigid motion in the plane; 2. The shapes in the set f퐶, (1,...,(=g have pairwise disjoint interiors; 3. The boundary of each (8 shares at least one point with the boundary of 퐶. 4. The boundary of 퐶 lies entirely within the interior of the union of 퐶 and all the (8. The second condition forces the shapes not to overlap, except on their boundaries. The third condition forces every (8 to be useful in covering the boundary of 퐶. The fourth condition ensures that 퐶 is completely surrounded. If, furthermore, the union of 퐶 and the (8 is simply connected, we say that 퐶 can be surrounded by ( without holes. In tiling theory, a finite union of non-overlapping shapes whose union is a topological disc is also known as a patch, a term I will use here. On the other hand, I will use the more general term packing when shapes are known to be non-overlapping but when their union may or may not contain holes. We formalize the notion of layers by defining the coronas of (. We define the 0-corona of ( to be the singleton set f(g. Setting 퐶 = ( above, if ( can be surrounded by itself then the tiles that make up that surround are one possible 1-corona of (. In general, if we have a nested sequence of :-coronas for : = 0, . , = − 1, all without holes, and the patch created from the union of all of these coronas can itself be surrounded by (, then the copies of ( making up the surround constitute an =-corona. The Heesch number of a shape ( is the largest = for which ( has an =-corona. If ( tiles the plane, then by definition it is possible to build an =-corona for every positive integer =, and we define its Heesch number to be infinity. If we wish to be concise, we will simply say that ( has 퐻 = =. 3 The definitions above require that for a shape to have Heesch number =, each :-corona for : = 1, . , =−1 surround its predecessor without holes. But it leaves the status of the outermost corona ambiguous. Most researchers require that a shape’s =-corona be hole-free in order to regard the shape as having 퐻 = =, but some permit the =-corona to have holes.
Recommended publications
  • Snakes in the Plane
    Snakes in the Plane by Paul Church A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Mathematics in Computer Science Waterloo, Ontario, Canada, 2008 c 2008 Paul Church I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract Recent developments in tiling theory, primarily in the study of anisohedral shapes, have been the product of exhaustive computer searches through various classes of poly- gons. I present a brief background of tiling theory and past work, with particular empha- sis on isohedral numbers, aperiodicity, Heesch numbers, criteria to characterize isohedral tilings, and various details that have arisen in past computer searches. I then develop and implement a new “boundary-based” technique, characterizing shapes as a sequence of characters representing unit length steps taken from a finite lan- guage of directions, to replace the “area-based” approaches of past work, which treated the Euclidean plane as a regular lattice of cells manipulated like a bitmap. The new technique allows me to reproduce and verify past results on polyforms (edge-to-edge as- semblies of unit squares, regular hexagons, or equilateral triangles) and then generalize to a new class of shapes dubbed polysnakes, which past approaches could not describe. My implementation enumerates polyforms using Redelmeier’s recursive generation algo- rithm, and enumerates polysnakes using a novel approach.
    [Show full text]
  • Polyominoes with Maximum Convex Hull Sascha Kurz
    University of Bayreuth Faculty for Mathematics and Physics Polyominoes with maximum convex hull Sascha Kurz Bayreuth, March 2004 Diploma thesis in mathematics advised by Prof. Dr. A. Kerber University of Bayreuth and by Prof. Dr. H. Harborth Technical University of Braunschweig Contents Contents . i List of Figures . ii List of Tables . iv 0 Preface vii Acknowledgments . viii Declaration . ix 1 Introduction 1 2 Proof of Theorem 1 5 3 Proof of Theorem 2 15 4 Proof of Theorem 3 21 5 Prospect 29 i ii CONTENTS References 30 Appendix 42 A Exact numbers of polyominoes 43 A.1 Number of square polyominoes . 44 A.2 Number of polyiamonds . 46 A.3 Number of polyhexes . 47 A.4 Number of Benzenoids . 48 A.5 Number of 3-dimensional polyominoes . 49 A.6 Number of polyominoes on Archimedean tessellations . 50 B Deutsche Zusammenfassung 57 Index 60 List of Figures 1.1 Polyominoes with at most 5 squares . 2 2.1 Increasing l1 ............................. 6 2.2 Increasing v1 ............................ 7 2.3 2-dimensional polyomino with maximum convex hull . 7 2.4 Increasing l1 in the 3-dimensional case . 8 3.1 The 2 shapes of polyominoes with maximum convex hull . 15 3.2 Forbidden sub-polyomino . 16 1 4.1 Polyominoes with n squares and area n + 2 of the convex hull . 22 4.2 Construction 1 . 22 4.3 Construction 2 . 23 4.4 m = 2n − 7 for 5 ≤ n ≤ 8 ..................... 23 4.5 Construction 3 . 23 iii iv LIST OF FIGURES 4.6 Construction 4 . 24 4.7 Construction 5 . 25 4.8 Construction 6 .
    [Show full text]
  • Open Problem Collection
    Open problem collection Peter Kagey May 14, 2021 This is a catalog of open problems that I began in late 2017 to keep tabs on different problems and ideas I had been thinking about. Each problem consists of an introduction, a figure which illustrates an example, a question, and a list related questions. Some problems also have references which refer to other problems, to the OEIS, or to other web references. 1 Rating Each problem is rated both in terms of how difficult and how interesting I think the problem is. 1.1 Difficulty The difficulty score follows the convention of ski trail difficulty ratings. Easiest The problem should be solvable with a modest amount of effort. Moderate Significant progress should be possible with moderate effort. Difficult Significant progress will be difficult or take substantial insight. Most difficult The problem may be intractable, but special cases may be solvable. 1.2 Interest The interest rating follows a four-point scale. Each roughly describes what quartile I think it belongs in with respect to my interest in it. Least interesting Problems have an interesting idea, but may feel contrived. More interesting Either a somewhat complicated or superficial question. Very interesting Problems that are particularly natural or simple or cute. Most interesting These are the problems that I care the most about. Open problem collection Peter Kagey Problem 1. Suppose you are given an n × m grid, and I then think of a rectangle with its corners on grid points. I then ask you to \black out" as many of the gridpoints as possible, in such a way that you can still guess my rectangle after I tell you all of the non-blacked out vertices that its corners lie on.
    [Show full text]
  • A Flowering of Mathematical Art
    A Flowering of Mathematical Art Jim Henle & Craig Kasper The Mathematical Intelligencer ISSN 0343-6993 Volume 42 Number 1 Math Intelligencer (2020) 42:36-40 DOI 10.1007/s00283-019-09945-0 1 23 Your article is protected by copyright and all rights are held exclusively by Springer Science+Business Media, LLC, part of Springer Nature. This e-offprint is for personal use only and shall not be self-archived in electronic repositories. If you wish to self- archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy For Our Mathematical Pleasure (Jim Henle, Editor) 1 have argued that the creation of mathematical A Flowering of structures is an art. The previous column discussed a II tiny genre of that art: numeration systems. You can’t describe that genre as ‘‘flowering.’’ But activity is most Mathematical Art definitely blossoming in another genre. Around the world, hundreds of artists are right now creating puzzles of JIM HENLE , AND CRAIG KASPER subtlety, depth, and charm. We are in the midst of a renaissance of logic puzzles. A Renaissance The flowering began with the discovery in 2004 in England, of the discovery in 1980 in Japan, of the invention in 1979 in the United States, of the puzzle type known today as sudoku.
    [Show full text]
  • Gemblo and Polyhex Blokus
    with John Gough < jagough49@gmail.com> Gemblo and Polyhex Blokus Blokus joined together whole-edge to whole-edge. Gemblo is one of the possible polyhex Blokus is an abstract strategy board game games that might be invented. Why for 2 to 4 players. It uses polyominoes obvious? Because both Blokus and Blokus WHicH are plane geometric lgures formed Trigon used families of shapes made from by joining one or more equal squares edge regular polygons that tessellate. How many to edge refer &igure . 0olyominoes HaVe are there? been described as being “a polyform whose cells are squaresv and they are classiled Gemblo according to how many cells they have. ie. number of cells. Wikipedia, 2017 March In 2005, Gemblo was created by Justin Oh 19). Polyominoes and other mathematically in Korea. It is an abstract strategy board patterned families of shapes were invented game with translucent, coloured pieces, by Solomon Golomb, and popularised by each of which is made up of one to lve hex- Martin Gardner. agons. The six colours are clear, red, yellow, green, blue, and purple. Despite the obvious similarities to Blokus, Oh has stated that when he was inventing Gemblo he did not know about Blokus. The name is based on the word ‘gem’, alluding to the gem-like colours used for the play- ing pieces and ‘blocking’—a central feature of playing. The size of the playing board depends on the number of people playing. Figure 1: Examples of polyominoes With players using a hexagonal shaped board that has sides of 8 unit-hexagons. Blokus—using squares—has a sister- game called Blokus Trigon that uses poly- Playing Gemblo iamonds, those families of shapes made s Gemblo can be played as a one- from unit-sided equilateral triangles joined person puzzle or solitaire, or by whole-edge to whole-edge.
    [Show full text]
  • A New Mathematical Model for Tiling Finite Regions of the Plane with Polyominoes
    Volume 15, Number 2, Pages 95{131 ISSN 1715-0868 A NEW MATHEMATICAL MODEL FOR TILING FINITE REGIONS OF THE PLANE WITH POLYOMINOES MARCUS R. GARVIE AND JOHN BURKARDT Abstract. We present a new mathematical model for tiling finite sub- 2 sets of Z using an arbitrary, but finite, collection of polyominoes. Unlike previous approaches that employ backtracking and other refinements of `brute-force' techniques, our method is based on a systematic algebraic approach, leading in most cases to an underdetermined system of linear equations to solve. The resulting linear system is a binary linear pro- gramming problem, which can be solved via direct solution techniques, or using well-known optimization routines. We illustrate our model with some numerical examples computed in MATLAB. Users can download, edit, and run the codes from http://people.sc.fsu.edu/~jburkardt/ m_src/polyominoes/polyominoes.html. For larger problems we solve the resulting binary linear programming problem with an optimization package such as CPLEX, GUROBI, or SCIP, before plotting solutions in MATLAB. 1. Introduction and motivation 2 Consider a planar square lattice Z . We refer to each unit square in the lattice, namely [~j − 1; ~j] × [~i − 1;~i], as a cell.A polyomino is a union of 2 a finite number of edge-connected cells in the lattice Z . We assume that the polyominoes are simply-connected. The order (or area) of a polyomino is the number of cells forming it. The polyominoes of order n are called n-ominoes and the cases for n = 1; 2; 3; 4; 5; 6; 7; 8 are named monominoes, dominoes, triominoes, tetrominoes, pentominoes, hexominoes, heptominoes, and octominoes, respectively.
    [Show full text]
  • Coverage Path Planning Using Reinforcement Learning-Based TSP for Htetran—A Polyabolo-Inspired Self-Reconfigurable Tiling Robot
    sensors Article Coverage Path Planning Using Reinforcement Learning-Based TSP for hTetran—A Polyabolo-Inspired Self-Reconfigurable Tiling Robot Anh Vu Le 1,2 , Prabakaran Veerajagadheswar 1, Phone Thiha Kyaw 3 , Mohan Rajesh Elara 1 and Nguyen Huu Khanh Nhan 2,∗ 1 ROAR Lab, Engineering Product Development, Singapore University of Technology and Design, Singapore 487372, Singapore; leanhvu@tdtu.edu.vn (A.V.L); prabakaran@sutd.edu.sg (P.V); rajeshelara@sutd.edu.sg (M.R.E.) 2 Optoelectronics Research Group, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam 3 Department of Mechatronic Engineering, Yangon Technological University, Insein 11101, Myanmar; mlsdphonethk@gmail.com * Correspondence: nguyenhuukhanhnhan@tdtu.edu.vn Abstract: One of the critical challenges in deploying the cleaning robots is the completion of covering the entire area. Current tiling robots for area coverage have fixed forms and are limited to cleaning only certain areas. The reconfigurable system is the creative answer to such an optimal coverage problem. The tiling robot’s goal enables the complete coverage of the entire area by reconfigur- ing to different shapes according to the area’s needs. In the particular sequencing of navigation, it is essential to have a structure that allows the robot to extend the coverage range while saving Citation: Le, A.V.; energy usage during navigation. This implies that the robot is able to cover larger areas entirely Veerajagadheswar, P.; Thiha Kyaw, P.; with the least required actions. This paper presents a complete path planning (CPP) for hTetran, Elara, M.R.; Nhan, N.H.K.
    [Show full text]
  • SUDOKU SOLVER Study and Implementation
    SUDOKU SOLVER Study and Implementation Abstract In this project we studied two algorithms to solve a Sudoku game and implemented a playable application for all major operating systems. Qizhong Mao, Baiyi Tao, Arif Aziz {qzmao, tud46571, arif.aziz}@temple.edu CIS 5603 Project Report Qizhong Mao, Baiyi Tao, Arif Aziz Table of Contents Introduction .......................................................................................................................... 3 Variants ................................................................................................................................ 3 Standard Sudoku ...........................................................................................................................3 Small Sudoku .................................................................................................................................4 Mini Sudoku ..................................................................................................................................4 Jigsaw Sudoku ...............................................................................................................................4 Hyper Sudoku ................................................................................................................................5 Dodeka Sudoku ..............................................................................................................................5 Number Place Challenger ...............................................................................................................5
    [Show full text]
  • Edgy Puzzles Karl Schaffer Karl Schaffer@Yahoo.Com
    Edgy Puzzles Karl Schaffer karl_schaffer@yahoo.com Countless puzzles involve decomposing areas or volumes of two or three-dimensional figures into smaller figures. “Polyform” puzzles include such well-known examples as pentominoes, tangrams, and soma cubes. This paper will examine puzzles in which the edge sets, or "skeletons," of various symmetric figures like polyhedra are decomposed into multiple copies of smaller graphs, and note their relationship to representations by props or body parts in dance performance. The edges of the tetrahedron in Figure 1 are composed of a folded 9-gon, while the cube and octahedron are each composed of six folded paths of length 2. These constructions have been used in dances created by the author and his collaborators. The photo from the author’s 1997 dance “Pipe Dreams” shows an octahedron in which each dancer wields three lengths of PVC pipe held together by cord at the two internal vertices labeled a in the diagram on the right. The shapes created by the dancers, which might include whimsical designs reminiscent of animals or other objects as well as mathematical forms, seem to appear and dissolve in fluid patterns, usually in time to a musical score. L R a R L a L R Figure 1. PVC pipe polyhedral skeletons used in dances The desire in the dance company co-directed by the author to incorporate polyhedra into dance works led to these constructions, and to similar designs with loops of rope, fingers and hands, and the bodies of dancers. Just as mathematical concepts often suggest artistic explorations for those involved in the interplay between these fields, performance problems may suggest mathematical questions, in this case involving finding efficient and symmetrical ways to construct the skeletons, or edge sets, of the Platonic solids.
    [Show full text]
  • Treb All De Fide Gra U
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by UPCommons. Portal del coneixement obert de la UPC Grau en Matematiques` T´ıtol:Tilings and the Aztec Diamond Theorem Autor: David Pardo Simon´ Director: Anna de Mier Departament: Mathematics Any academic:` 2015-2016 TREBALL DE FI DE GRAU Facultat de Matemàtiques i Estadística David Pardo 2 Tilings and the Aztec Diamond Theorem A dissertation submitted to the Polytechnic University of Catalonia in accordance with the requirements of the Bachelor's degree in Mathematics in the School of Mathematics and Statistics. David Pardo Sim´on Supervised by Dr. Anna de Mier School of Mathematics and Statistics June 28, 2016 Abstract Tilings over the plane R2 are analysed in this work, making a special focus on the Aztec Diamond Theorem. A review of the most relevant results about monohedral tilings is made to continue later by introducing domino tilings over subsets of R2. Based on previous work made by other mathematicians, a proof of the Aztec Dia- mond Theorem is presented in full detail by completing the description of a bijection that was not made explicit in the original work. MSC2010: 05B45, 52C20, 05A19. iii Contents 1 Tilings and basic notions1 1.1 Monohedral tilings............................3 1.2 The case of the heptiamonds.......................8 1.2.1 Domino Tilings.......................... 13 2 The Aztec Diamond Theorem 15 2.1 Schr¨odernumbers and Hankel matrices................. 16 2.2 Bijection between tilings and paths................... 19 2.3 Hankel matrices and n-tuples of Schr¨oderpaths............ 27 v Chapter 1 Tilings and basic notions The history of tilings and patterns goes back thousands of years in time.
    [Show full text]
  • Jigsaw Puzzles, Edge Matching, and Polyomino Packing: Connections and Complexity∗
    Jigsaw Puzzles, Edge Matching, and Polyomino Packing: Connections and Complexity∗ Erik D. Demaine, Martin L. Demaine MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar St., Cambridge, MA 02139, USA, {edemaine,mdemaine}@mit.edu Dedicated to Jin Akiyama in honor of his 60th birthday. Abstract. We show that jigsaw puzzles, edge-matching puzzles, and polyomino packing puzzles are all NP-complete. Furthermore, we show direct equivalences between these three types of puzzles: any puzzle of one type can be converted into an equivalent puzzle of any other type. 1. Introduction Jigsaw puzzles [37,38] are perhaps the most popular form of puzzle. The original jigsaw puzzles of the 1760s were cut from wood sheets using a hand woodworking tool called a jig saw, which is where the puzzles get their name. The images on the puzzles were European maps, and the jigsaw puzzles were used as educational toys, an idea still used in some schools today. Handmade wooden jigsaw puzzles for adults took off around 1900. Today, jigsaw puzzles are usually cut from cardboard with a die, a technology that became practical in the 1930s. Nonetheless, true addicts can still find craftsmen who hand-make wooden jigsaw puzzles. Most jigsaw puzzles have a guiding image and each side of a piece has only one “mate”, although a few harder variations have blank pieces and/or pieces with ambiguous mates. Edge-matching puzzles [21] are another popular puzzle with a similar spirit to jigsaw puzzles, first appearing in the 1890s. In an edge-matching puzzle, the goal is to arrange a given collection of several identically shaped but differently patterned tiles (typically squares) so that the patterns match up along the edges of adjacent tiles.
    [Show full text]
  • Formulae and Growth Rates of Animals on Cubical and Triangular Lattices
    Formulae and Growth Rates of Animals on Cubical and Triangular Lattices Mira Shalah Technion - Computer Science Department - Ph.D. Thesis PHD-2017-18 - 2017 Technion - Computer Science Department - Ph.D. Thesis PHD-2017-18 - 2017 Formulae and Growth Rates of Animals on Cubical and Triangular Lattices Research Thesis Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Mira Shalah Submitted to the Senate of the Technion | Israel Institute of Technology Elul 5777 Haifa September 2017 Technion - Computer Science Department - Ph.D. Thesis PHD-2017-18 - 2017 Technion - Computer Science Department - Ph.D. Thesis PHD-2017-18 - 2017 This research was carried out under the supervision of Prof. Gill Barequet, at the Faculty of Computer Science. Some results in this thesis have been published as articles by the author and research collaborators in conferences and journals during the course of the author's doctoral research period, the most up-to-date versions of which being: • Gill Barequet and Mira Shalah. Polyominoes on Twisted Cylinders. Video Review at the 29th Symposium on Computational Geometry, 339{340, 2013. https://youtu.be/MZcd1Uy4iv8. • Gill Barequet and Mira Shalah. Automatic Proofs for Formulae Enu- merating Proper Polycubes. In Proceedings of the 8th European Conference on Combinatorics, Graph Theory and Applications, 145{151, 2015. Also in Video Review at the 31st Symposium on Computational Geometry, 19{22, 2015. https://youtu.be/ojNDm8qKr9A. Full version: Gill Barequet and Mira Shalah. Counting n-cell Polycubes Proper in n − k Dimensions. European Journal of Combinatorics, 63 (2017), 146{163. • Gill Barequet, Gunter¨ Rote and Mira Shalah.
    [Show full text]